Singularity formations in non linear partial differential equations

In many non-linear evolutionary or stationary partial differential equations (PDEs), one observes the formation of singularities or some form of concentration of their solutions, as the time-variable or a parameter of the model approaches a limit value. In that case solutions become highly concentrated on lower-dimensional sets, thus losing smoothness and approaching a singular limit. This project is devoted to the study of formation of singularities for solutions of classes of non-linear PDEs. The questions the project intend to answer are: Do singularities occur? What is the mechanism that triggers the formation of singularities? Where (in space) and when (in time) do singularities develop? What is the shape of such singularities? What happens after the formation of singularities?

The novel mathematical methodology that will be carried out during the project consists of identifying first the form and location of a possible singularity and using this to construct a good approximate solution. This first step requires a deep understanding of the model the PDEs are describing. The second step consists of designing an analytic strategy to produce an actual solution rather than an approximate one, for instance by using a perturbation argument. The student will consider some model PDEs which describe the motion of an incompressible fluid in dimension two, such as Euler equations, Navier-Stokes equations and the lake equations, as well as some parabolic critical non-linear PDEs. The initial aim is to construct solutions with bounded initial vorticity which produce a global solution whose gradient grows in time as a double exponential for the lake equation, using as a reference the paper 'Small scale creation for solutions of the incompressible two dimensional Euler equation' by Kiselev and Sverak. The plan is also to investigate the evolution of concentrated vorticities in the Navier-Stokes 2-dimensional model for small viscosity, when the initial vorticity is highly concentrated around a given number of points. The specific aim is to build such solutions using gluing techniques.